The number of arrangements, of the letters of the word MANAMA in which two M's do not appear adjacent, is

A random variable $X$ has the following probability distribution

For the event $E=\{X$ is a prime number $\}$, $F=\{X<4\}$, then $P(E \cup F)$ is

$$\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}=$$

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half of the quantity of ice melts in 15 minutes. $x_0$ is the initial quantity of ice. If after 30 minutes the amount of ice left is $\mathrm{kx}_0$, then the value of $k$ is